数学 - Python - 代数学 - 因数分解と分数式 - 因数分解 - 雑例 - 様々な工夫

学習環境

代数への出発 (新装版数学入門シリーズ) (松坂和夫(著)、岩波書店)の第3章(因数分解と分数式)、1(因数分解)の問9の解答を求めてみる。

1. $\begin{array}{l} a^{4} - b^{2} c^{2} + a^{2} c^{2} - b^{4} \\ = (a^{2} - b^{2}) (a^{2} + b^{2}) + c^{2} (a^{2} - b^{2}) \\ = (a + b) (a - b) (a^{2} + b^{2} + c^{2}) \end{array}$
2. $\begin{array}{l} x^{3} - y^{3} - x^{2} + 2 x y - y^{2} \\ = (x - y) (x^{2} + x y + y^{2}) - {(x - y)}^{2} \\ = (x - y) (x^{2} + x y + y^{2} - x + y) \end{array}$
3. $\begin{array}{l} a^{4} + 2 a^{3} b - 2 a b^{3} - b^{4} \\ = (a^{2} - b^{2}) (a^{2} + b^{2}) + 2 a b (a^{2} - b^{2}) \\ = (a + b) (a - b) {(a + b)}^{2} \\ = {(a + b)}^{3} (a - b) \end{array}$
4. $\begin{array}{l} 3 x^{3} + x^{2} y + 2 y^{3} \\ = 3 x^{3} + 3 y^{3} - y^{3} + x^{2} y \\ = 3 (x + y) (x^{2} - x y + y^{2}) - y (y^{2} - x^{2}) \\ = 3 (x + y) (x^{2} - x y + y^{2}) + y (x + y) (x - y) \\ = (x + y) (3 x^{2} - 3 x y + 3 y^{2} + x y - y^{2}) \\ = (x + y) (3 x^{2} - 2 x y + 2 y^{2}) \end{array}$
5. $\begin{array}{l} a^{3} + 3 a b^{2} + 4 b^{3} \\ = a^{3} + b^{3} + 3 b^{3} + 3 a b^{2} \\ = (a + b) (a^{2} - a b + b^{2}) + 3 b^{2} (a + b) \\ = (a + b) (a^{2} - a b + 4 b^{2}) \end{array}$
6. $\begin{array}{l} {(a + b + c)}^{3} - a^{3} - b^{3} - c^{3} \\ = a^{3} + 3 a^{2} (b + c) + 3 a {(b + c)}^{2} + {(b + c)}^{3} \\ - a^{3} - b^{3} - c^{3} \\ = 3 a^{2} (b + c) + 3 a {(b + c)}^{2} + 3 b^{2} c + 3 b c^{2} \\ = 3 a^{2} (b + c) + 3 a {(b + c)}^{2} + 3 b c (b + c) \\ = 3 (b + c) (a^{2} + a (b + c) + b c) \\ = 3 (b + c) (a + b) (a + c) \\ = 3 (a + b) (b + c) (c + a) \end{array}$
7. $\begin{array}{l} (x^{2} + x - 2) (x^{2} + x + 1) - 28 \\ = {(x^{2} + x)}^{2} - (x^{2} + x) - 30 \\ = (x^{2} + x - 6) (x^{2} + x + 5) \\ = (x + 3) (x - 2) (x^{2} + x + 5) \end{array}$
8. $\begin{array}{l} (2 x^{2} - 3 x) (2 x^{2} - 3 x - 6) + 5 \\ = {(2 x^{2} - 3 x)}^{2} - 6 (2 x^{2} - 3 x) + 5 \\ = (2 x^{2} - 3 x - 1) (2 x^{2} - 3 x - 5) \\ = (2 x^{2} - 3 x - 1) (2 x - 5) (x + 1) \end{array}$

#!/usr/bin/env python3 from unittest import TestCase, main from sympy import symbols print('9.') class MyTest(TestCase): def test1(self): x, y, z, a, b, c = symbols('x, y, z, a, b, c', real=True) spam = [a ** 4 - b ** 2 * c ** 2 + a ** 2 * c ** 2 - b ** 4, (x ** 3 - y ** 3 - x ** 2 + 2 * x * y - y ** 2), a ** 4 + 2 * a ** 3 * b - 2 * a * b ** 3 - b ** 4, 3 * x ** 3 + x ** 2 * y + 2 * y ** 3, a ** 3 + 3 * a * b ** 2 + 4 * b ** 3, (a + b + c) ** 3 - a ** 3 - b ** 3 - c ** 3, (x ** 2 + x - 2) * (x ** 2 + x + 1) - 28, (2 * x ** 2 - 3 * x) * (2 * x ** 2 - 3 * x - 6) + 5] egg = [(a + b) * (a - b) * (a ** 2 + b ** 2 + c ** 2), (x - y) * (x ** 2 + x * y + y ** 2 - x + y), (a + b) ** 3 * (a - b), (x + y) * (3 * x ** 2 - 2 * x * y + 2 * y ** 2), (a + b) * (a ** 2 - a * b + 4 * b ** 2), 3 * (a + b) * (b + c) * (c + a), (x + 3) * (x - 2) * (x ** 2 + x + 5), (2 * x ** 2 - 3 * x - 1) * (2 * x - 5) * (x + 1)] for s, t in zip(spam, egg): self.assertEqual(s.expand(), t.expand()) self.assertEqual(s.factor(), t.factor()) if __name__ == '__main__': main()

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2019年12月31日火曜日

数学 - Python - 代数学 - 因数分解と分数式 - 因数分解 - 雑例 - 様々な工夫

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